Making #z = 2 e^(i phi)# we have
#z*cos(z/(z+1))dz = 2e^(i phi) (sum_(k=0)^oo (-1)^k/(2k!)((2e^(i phi))/(2e^(i phi)+1))^(2k))2i e^(i phi) d phi#
and now
#int_(|z|=2)z*cos(z/(z+1))dz = int_0^(2pi) 2e^(i phi) (sum_(k=0)^oo (-1)^k/(2k!)((2e^(i phi))/(2e^(i phi)+1))^(2k))2i e^(i phi) d phi = sum_(k=0)^oo a_k#
where
#a_k = 4i int_0^(2pi)(-1)^k/(2k!)((2e^(i phi))/(2e^(i phi)+1))^(2k) e^(i phi) d phi #
and then
#a_0 = 0#
#a_1 = -3ipi#
#a_2 = (5i pi)/6#
#a_3=-(7ipi)/120#
#a_4 =(i pi)/560#
#a_5=-(11i pi)/(362880)#
#a_6 = (13 i pi)/39916800#
#cdots#
which converges quickly .
And finally
#int_(|z|=2)z*cos(z/(z+1))dz approx -(9860533 i pi)/4435200 = -2.22324 i pi#
